One could then proceed by contracting \((\ref {Pt})\)  with \(u^{\mu }u^{\nu }\)  in order to eliminate the first term (\(R_{\mu \nu }u^{\mu }u^{\nu }\) ) and express everything in terms of the scale factor and the torsion and non-metricity variables. This results in a fairly complicated expression which we refrain from presenting it here since it goes beyond the scope of the present exposure. As a final note let us mention that in order to analyse in depth the above cosmological model one should consider an appropriate form of matter for which both the metrical energy momentum and hypermomentum tensors must respect the cosmological principle. The fluid with such characteristics was constructed in [1]} (see also for a generalized version [2]}) and goes by the name Perfect Cosmological Hyperfluid. The hypermomentum part of this fluid will then source the torsion and non-metricity variables \(\Phi , P,A,...\)  etc. by virtue of the connection field equations. We note that scalar fields coupled to the connection belong (are certain subcases) to the aforementioned fluid description. For the sake of illustration, below we present such an example with a scalar field non-minimally coupled to the connection in the case of vanishing non-metricity and also study some of the cosmological implications of this theory.
